Appendix A: The conditions for an OC

 In this appendix, we show that conditions 8 diagonalise the error matrix $\bf{V}$ of the Stokes parameters and minimise its determinant, if the noises in the n polarimeters have identical levels and are not correlated.
We use the notation: It can be seen from Eq. 7 that requiring that the error on I be decorrelated from the errors on Q and U is equivalent to the condition:

 

S₂ = 0. (A1)

This condition can easily be fulfilled in a configuration where the angles $\alpha_{\rm p}$ are regularly distributed:

$$\begin{eqnarray} & \alpha_{ p} = \alpha_1 + (p-1)\,\delta\alpha,\ p = 1...\ n,\\... ...lpha < \pi, \ \delta\alpha \ne \pi/2\ \mbox{ (see text)}.\nonumber\end{eqnarray}$$ (A2)

In such configurations, Eq. (A1) becomes:

$$\begin{eqnarray} S_2 = {\rm e}^{i\,2\,\alpha_1}\frac{{\rm e}^{i\,2\,n\,\delta\alpha}-1}{{\rm e}^{i\,2\,\delta\alpha}-1} = 0.\end{eqnarray}$$ (A3)

The solutions of Eq. (A3) under conditions (A2) reduce to:

$$\begin{eqnarray} \delta\alpha = \frac{\pi}{n}, \mbox{ with } n \ge 3.\end{eqnarray}$$ (A4)

It is easily seen that conditions (A4) also automatically ensure that S₄ = 0 and therefore that $\bf{X}$ becomes diagonal and assumes the very simple form:

$$\begin{eqnarray} {\bf X}_0 = \frac{n}{4} \left(\begin{array} {ccc} 1 & 0 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/2 \\ \end{array}\right)\end{eqnarray}$$ (A5)

independent of the orientation of the focal plane. Equation 9 is the consequence of A5.

The error volume is proportional to the determinant of the error matrix $\bf{V}$. Therefore, it is minimum when the determinant of $\bf{X}$ (Eq. 7) is maximum. This determinant can be written as:  

$$\mathrm{Det}(\bf{X}) =\\ \frac{1}{64} \left[n^3 - n\,\vert ... ...^2(n - \vert S_4\vert\, \cos(\theta_4 - 2\, \theta_2))\right].$$ (A6)

Because the S_k's are sums of n complex numbers with modulus 1, |S_k| < n, and it is clear from Eq. A6 that and that the upper bound is reached if and only if

 

S₂ = S₄ = 0. (A7)

Conditions (A2) and A4 have been shown above to imply A7, and therefore ensure that the determinant of the covariance matrix $\bf{V}$ is minimum.